Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [102,4,Mod(19,102)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(102, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 7]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("102.19");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 102 = 2 \cdot 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 102.h (of order \(8\), degree \(4\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(6.01819482059\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{8})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 184 x^{14} - 184 x^{13} + 19194 x^{12} + 27184 x^{11} + 973936 x^{10} + 4442568 x^{9} + \cdots + 138976206259489 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{16} + 184 x^{14} - 184 x^{13} + 19194 x^{12} + 27184 x^{11} + 973936 x^{10} + 4442568 x^{9} + \cdots + 138976206259489 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 13\!\cdots\!03 \nu^{15} + \cdots - 64\!\cdots\!45 ) / 18\!\cdots\!36 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14\!\cdots\!15 \nu^{15} + \cdots + 28\!\cdots\!09 ) / 18\!\cdots\!36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 70\!\cdots\!10 \nu^{15} + \cdots - 15\!\cdots\!44 ) / 62\!\cdots\!88 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 47\!\cdots\!06 \nu^{15} + \cdots - 31\!\cdots\!85 ) / 39\!\cdots\!76 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 48\!\cdots\!20 \nu^{15} + \cdots + 76\!\cdots\!48 ) / 39\!\cdots\!76 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 10\!\cdots\!11 \nu^{15} + \cdots - 92\!\cdots\!65 ) / 55\!\cdots\!68 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 10\!\cdots\!53 \nu^{15} + \cdots - 35\!\cdots\!71 ) / 39\!\cdots\!76 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 11\!\cdots\!83 \nu^{15} + \cdots - 38\!\cdots\!25 ) / 39\!\cdots\!76 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 12\!\cdots\!69 \nu^{15} + \cdots + 23\!\cdots\!68 ) / 39\!\cdots\!76 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 22\!\cdots\!42 \nu^{15} + \cdots + 72\!\cdots\!13 ) / 55\!\cdots\!68 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 20\!\cdots\!16 \nu^{15} + \cdots + 62\!\cdots\!87 ) / 39\!\cdots\!76 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 27\!\cdots\!81 \nu^{15} + \cdots + 84\!\cdots\!31 ) / 27\!\cdots\!84 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 48\!\cdots\!99 \nu^{15} + \cdots - 84\!\cdots\!32 ) / 39\!\cdots\!76 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 66\!\cdots\!96 \nu^{15} + \cdots + 16\!\cdots\!57 ) / 39\!\cdots\!76 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 40\!\cdots\!03 \nu^{15} + \cdots - 93\!\cdots\!37 ) / 19\!\cdots\!88 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{15} + 3\beta_{13} - 3\beta_{10} + 3\beta_{7} - \beta_{5} + 3\beta_{3} - 3 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3 \beta_{14} - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{9} - 68 \beta_{5} + 5 \beta_{4} + 30 \beta_{3} + \cdots - 69 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 57 \beta_{15} - 42 \beta_{14} - 99 \beta_{13} + 30 \beta_{12} - 42 \beta_{11} + 50 \beta_{9} + \cdots + 138 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 609 \beta_{15} - 633 \beta_{14} - 333 \beta_{13} - 609 \beta_{11} - 225 \beta_{10} + 3049 \beta_{9} + \cdots - 1962 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6102 \beta_{15} + 4527 \beta_{14} + 4527 \beta_{12} - 7209 \beta_{11} + 18432 \beta_{10} + 3561 \beta_{9} + \cdots - 44958 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 100368 \beta_{15} - 40815 \beta_{13} + 100368 \beta_{12} + 6771 \beta_{11} + 88248 \beta_{10} + \cdots + 672501 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 516336 \beta_{15} + 516336 \beta_{14} + 1414662 \beta_{13} - 688167 \beta_{12} + 2441883 \beta_{11} + \cdots + 3223524 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 14107758 \beta_{14} + 8229414 \beta_{13} - 13170942 \beta_{12} + 6838794 \beta_{11} - 936816 \beta_{10} + \cdots - 4810017 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 51390075 \beta_{15} - 67966740 \beta_{14} - 141966987 \beta_{13} - 68636490 \beta_{12} + \cdots + 537025563 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 615412254 \beta_{15} - 566609937 \beta_{14} + 45826212 \beta_{13} - 2976105 \beta_{12} + \cdots - 2285462315 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 5805023829 \beta_{15} - 6383301888 \beta_{14} + 3170698845 \beta_{13} + 20243391726 \beta_{12} + \cdots - 128178357366 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 208716066225 \beta_{15} - 8754331701 \beta_{14} - 208716066225 \beta_{13} + 135421068486 \beta_{12} + \cdots + 350599408032 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 598533041880 \beta_{15} + 1681947979221 \beta_{14} + 106411678680 \beta_{13} - 1681947979221 \beta_{12} + \cdots + 6083692817550 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 1823987296242 \beta_{15} + 17225249668704 \beta_{14} + 24653876872827 \beta_{13} - 1823987296242 \beta_{12} + \cdots + 59929388360265 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 123249163418262 \beta_{15} - 123249163418262 \beta_{14} + 37376027471520 \beta_{13} + \cdots + 986415807473232 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).
| \(n\) | \(35\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
1.41421 | + | 1.41421i | −2.77164 | + | 1.14805i | 4.00000i | −7.23380 | − | 17.4639i | −5.54328 | − | 2.29610i | 0.436834 | − | 1.05461i | −5.65685 | + | 5.65685i | 6.36396 | − | 6.36396i | 14.4676 | − | 34.9279i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.2 | 1.41421 | + | 1.41421i | −2.77164 | + | 1.14805i | 4.00000i | 4.50563 | + | 10.8776i | −5.54328 | − | 2.29610i | −0.303458 | + | 0.732613i | −5.65685 | + | 5.65685i | 6.36396 | − | 6.36396i | −9.01126 | + | 21.7551i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.3 | 1.41421 | + | 1.41421i | 2.77164 | − | 1.14805i | 4.00000i | −5.79363 | − | 13.9871i | 5.54328 | + | 2.29610i | 12.9100 | − | 31.1676i | −5.65685 | + | 5.65685i | 6.36396 | − | 6.36396i | 11.5873 | − | 27.9741i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.4 | 1.41421 | + | 1.41421i | 2.77164 | − | 1.14805i | 4.00000i | 4.27917 | + | 10.3308i | 5.54328 | + | 2.29610i | −3.14393 | + | 7.59011i | −5.65685 | + | 5.65685i | 6.36396 | − | 6.36396i | −8.55834 | + | 20.6616i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.1 | −1.41421 | + | 1.41421i | −1.14805 | + | 2.77164i | − | 4.00000i | −2.14284 | − | 0.887595i | −2.29610 | − | 5.54328i | −10.9809 | + | 4.54844i | 5.65685 | + | 5.65685i | −6.36396 | − | 6.36396i | 4.28569 | − | 1.77519i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.2 | −1.41421 | + | 1.41421i | −1.14805 | + | 2.77164i | − | 4.00000i | 8.34237 | + | 3.45552i | −2.29610 | − | 5.54328i | 20.2720 | − | 8.39695i | 5.65685 | + | 5.65685i | −6.36396 | − | 6.36396i | −16.6847 | + | 6.91104i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.3 | −1.41421 | + | 1.41421i | 1.14805 | − | 2.77164i | − | 4.00000i | −2.90099 | − | 1.20163i | 2.29610 | + | 5.54328i | −22.4306 | + | 9.29105i | 5.65685 | + | 5.65685i | −6.36396 | − | 6.36396i | 5.80197 | − | 2.40325i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.4 | −1.41421 | + | 1.41421i | 1.14805 | − | 2.77164i | − | 4.00000i | 0.944104 | + | 0.391061i | 2.29610 | + | 5.54328i | 3.23995 | − | 1.34203i | 5.65685 | + | 5.65685i | −6.36396 | − | 6.36396i | −1.88821 | + | 0.782121i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.1 | 1.41421 | − | 1.41421i | −2.77164 | − | 1.14805i | − | 4.00000i | −7.23380 | + | 17.4639i | −5.54328 | + | 2.29610i | 0.436834 | + | 1.05461i | −5.65685 | − | 5.65685i | 6.36396 | + | 6.36396i | 14.4676 | + | 34.9279i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | 1.41421 | − | 1.41421i | −2.77164 | − | 1.14805i | − | 4.00000i | 4.50563 | − | 10.8776i | −5.54328 | + | 2.29610i | −0.303458 | − | 0.732613i | −5.65685 | − | 5.65685i | 6.36396 | + | 6.36396i | −9.01126 | − | 21.7551i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | 1.41421 | − | 1.41421i | 2.77164 | + | 1.14805i | − | 4.00000i | −5.79363 | + | 13.9871i | 5.54328 | − | 2.29610i | 12.9100 | + | 31.1676i | −5.65685 | − | 5.65685i | 6.36396 | + | 6.36396i | 11.5873 | + | 27.9741i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.4 | 1.41421 | − | 1.41421i | 2.77164 | + | 1.14805i | − | 4.00000i | 4.27917 | − | 10.3308i | 5.54328 | − | 2.29610i | −3.14393 | − | 7.59011i | −5.65685 | − | 5.65685i | 6.36396 | + | 6.36396i | −8.55834 | − | 20.6616i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.1 | −1.41421 | − | 1.41421i | −1.14805 | − | 2.77164i | 4.00000i | −2.14284 | + | 0.887595i | −2.29610 | + | 5.54328i | −10.9809 | − | 4.54844i | 5.65685 | − | 5.65685i | −6.36396 | + | 6.36396i | 4.28569 | + | 1.77519i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | −1.41421 | − | 1.41421i | −1.14805 | − | 2.77164i | 4.00000i | 8.34237 | − | 3.45552i | −2.29610 | + | 5.54328i | 20.2720 | + | 8.39695i | 5.65685 | − | 5.65685i | −6.36396 | + | 6.36396i | −16.6847 | − | 6.91104i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | −1.41421 | − | 1.41421i | 1.14805 | + | 2.77164i | 4.00000i | −2.90099 | + | 1.20163i | 2.29610 | − | 5.54328i | −22.4306 | − | 9.29105i | 5.65685 | − | 5.65685i | −6.36396 | + | 6.36396i | 5.80197 | + | 2.40325i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | −1.41421 | − | 1.41421i | 1.14805 | + | 2.77164i | 4.00000i | 0.944104 | − | 0.391061i | 2.29610 | − | 5.54328i | 3.23995 | + | 1.34203i | 5.65685 | − | 5.65685i | −6.36396 | + | 6.36396i | −1.88821 | − | 0.782121i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.d | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 102.4.h.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | 306.4.l.c | 16 | ||
| 17.d | even | 8 | 1 | inner | 102.4.h.b | ✓ | 16 |
| 17.e | odd | 16 | 1 | 1734.4.a.bh | 8 | ||
| 17.e | odd | 16 | 1 | 1734.4.a.bi | 8 | ||
| 51.g | odd | 8 | 1 | 306.4.l.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.4.h.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 102.4.h.b | ✓ | 16 | 17.d | even | 8 | 1 | inner |
| 306.4.l.c | 16 | 3.b | odd | 2 | 1 | ||
| 306.4.l.c | 16 | 51.g | odd | 8 | 1 | ||
| 1734.4.a.bh | 8 | 17.e | odd | 16 | 1 | ||
| 1734.4.a.bi | 8 | 17.e | odd | 16 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} + 532 T_{5}^{14} - 4608 T_{5}^{13} + 141512 T_{5}^{12} - 1404728 T_{5}^{11} + \cdots + 6410851825156 \)
acting on \(S_{4}^{\mathrm{new}}(102, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 16)^{4} \)
|
| $3$ |
\( (T^{8} + 6561)^{2} \)
|
| $5$ |
\( T^{16} + \cdots + 6410851825156 \)
|
| $7$ |
\( T^{16} + \cdots + 31032653924416 \)
|
| $11$ |
\( T^{16} + \cdots + 22\!\cdots\!96 \)
|
| $13$ |
\( T^{16} + \cdots + 66\!\cdots\!56 \)
|
| $17$ |
\( T^{16} + \cdots + 33\!\cdots\!21 \)
|
| $19$ |
\( T^{16} + \cdots + 58\!\cdots\!84 \)
|
| $23$ |
\( T^{16} + \cdots + 31\!\cdots\!36 \)
|
| $29$ |
\( T^{16} + \cdots + 42\!\cdots\!84 \)
|
| $31$ |
\( T^{16} + \cdots + 24\!\cdots\!24 \)
|
| $37$ |
\( T^{16} + \cdots + 35\!\cdots\!96 \)
|
| $41$ |
\( T^{16} + \cdots + 26\!\cdots\!76 \)
|
| $43$ |
\( T^{16} + \cdots + 77\!\cdots\!84 \)
|
| $47$ |
\( T^{16} + \cdots + 30\!\cdots\!44 \)
|
| $53$ |
\( T^{16} + \cdots + 13\!\cdots\!64 \)
|
| $59$ |
\( T^{16} + \cdots + 31\!\cdots\!04 \)
|
| $61$ |
\( T^{16} + \cdots + 10\!\cdots\!36 \)
|
| $67$ |
\( (T^{8} + \cdots - 12\!\cdots\!68)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 35\!\cdots\!24 \)
|
| $73$ |
\( T^{16} + \cdots + 41\!\cdots\!24 \)
|
| $79$ |
\( T^{16} + \cdots + 11\!\cdots\!56 \)
|
| $83$ |
\( T^{16} + \cdots + 96\!\cdots\!84 \)
|
| $89$ |
\( T^{16} + \cdots + 31\!\cdots\!64 \)
|
| $97$ |
\( T^{16} + \cdots + 56\!\cdots\!96 \)
|
| show more | |
| show less |